Continued Fractions & Rational Approximations

Continued fractions can be thought of as an alternative to digit sequences for representing numbers, based on division rather than multiplication by a base. Studied occasionally for at least half a millennium, continued fractions have become increasingly important through their applications to dynamical systems theory and number theoretic algorithms. The Wolfram Language has highly efficient original algorithms for finding large numbers of terms in continued fractions, as well as for handling exact continued fractions for quadratic irrationals.

ContinuedFraction continued fraction expansion

FromContinuedFraction construct exact or inexact numbers from continued fractions

Convergents a list of successive convergents of a continued fraction

Rationalize find rational approximations

QuadraticIrrationalQ test for a quadratic irrational (repeating continued fraction)

ContinuedFractionK construct a continued fraction from a symbolic formula

Khinchin Khinchin's constant characterizing random continued fractions

Objects with Notable Continued Fractions

Sqrt  ▪  E  ▪  Tan  ▪  BesselI  ▪  Pi  ▪  ChampernowneNumber